Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 48973
Applying the Pythagorean Theorem
This lesson plan is lesson 1 of two lessons. This lesson applies the Pythagorean Theorem and teaches the foundational skills required to proceed to
lesson 2, Origami Boats - Pythagorean Theorem in the real world Resource ID 49055. This lesson should not be taught until the students have a
knowledge of standard MAFS.8.G.2.6 Explain a proof of the Pythagorean Theorem and its converse.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Suggested Technology: Computer for Presenter,
Basic Calculators, LCD Projector
Instructional Time: 1 Hour(s) 30 Minute(s)
Freely Available: Yes
Keywords: right triangle, triangle, squared, hypotenuse, Pythagoras, Pythagorean Theorem, missing length,
missing side, apply Pythagorean Theorem, unknown side lengths, leg
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
Pythagorean Theorem Worksheet.doc
Pythagorean Theorem Worksheet Answer.doc
Pythagorean Theorem QUIZ.doc
Pythagorean Theorem PowerPoint CPALMS.ppt
Pythagorean Theorem PowerPoint Accomodations CPALMS.ppt
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will apply the Pythagorean Theorem using it to find the missing length of a triangle.
Students will be able to:
Identify and apply the Pythagorean Theorem to find the missing hypotenuse of a triangle.
Identify and apply the Pythagorean Theorem to find the missing leg of a triangle
Look for and regularity in repeated reasoning.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should already have an understanding on MAFS.8.G.2.6 Explain a proof of the Pythagorean Theorem and its converse.
Students should be able to solve two-step equations.
Students should be able to calculate and estimate square roots.
Students should be able to evaluate expressions or equations with single digit exponents.
Vocabulary: right triangle, right angle, hypotenuse, legs, square root
Guiding Questions: What are the guiding questions for this lesson?
Display these questions for students to see while they work. Discuss them at the beginning, middle and end of the lesson.
page 1 of 5 What is the purpose of the Pythagorean Theorem? To find the missing length of the side of a right triangle
How can I use the Pythagorean Theorem to find the missing length of a right triangle? Use the formula
How does the length of the hypotenuse compare to the length of a leg? It is longer.
How can you tell if your answer for the length of a leg or hypotenuse is a reasonable answer? How does it compare to the other lengths, and is the answer within a
reasonable expectation?
Teaching Phase: How will the teacher present the concept or skill to students?
Use the Pythagorean Theorem PowerPoint to teach the lesson. Read the following before using the PowerPoint to help expand upon what is in the slides as you
present it. If a projector is unavailable, do the following.
1. Start off with an appropriate bell ringer(s) to assess the students' prior knowledge. A bell ringer is a problem that is prominently displayed in the classroom that the
students are required to work on once they enter the classroom. This is usually set up at the beginning of the year as a procedure. These bell ringers can be
checked as a separate assignment weekly or monthly, or they can be included in a notebook check. The numbers of questions may vary according to timing and the
level of your students. Some suggestions of appropriate bell ringers appear in the formative assessment section of this lesson.
2. Briefly review the vocabulary below. Ask students to write definitions or examples for each term on their own. This could be included as part of the bell ringers.
Have volunteers share their ideas and record/display their definitions. Make them part of a word wall.
triangle
right triangle
hypotenuse
legs
square root
1. Have a picture of a 3-4-5 right triangle on the board or display using an overhead with squares on the edges (This can also be done using the Pythagorean
Theorem Lesson PowerPoint, the picture is on slide 4).
Point out the right triangle to the students and ask them what the length of the bottom edge is. (3 units.)
Questions:
If the length of the bottom edge is 3 units, what do the 9 squares on the bottom here represent? (the area of a square with a length of 3, 3 squared)
What is the length of the left side of the triangle? (4, if they answer 16, point out that 16 is the area of the square, not the edge)
What is the length of the long side? (5, if they answer 16, point out that 16 is the area of the square, not the edge)
Is there a special name for this longer side? (hypotenuse)
Does there appear to be any relationship between the areas of these three squares and the triangle? What about between the squares themselves?
If I add the squares of the two shorter sides together, what do I get? (25, the area of the largest square connected to the hypoteneuse)
Why does that sum seem familiar? (it's the same as the square of the length of the hypotenuse.)
Some guy back in about 500 BC proved mathematically that there is a relationship here, and that it is true for all right triangles. This was roughly 800 years before
Algebra was established. This guy was the Greek philosopher Pythagoras of Samos. Ever since, his name has been associated with this theorem. Technically,
someone else figured this whole thing out hundreds of years before him.
What this theorem states is that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. a2+b2=c2
Let's try it with this example. 9 + 16 = 25
What can be done with this information? (find a missing length)
What if you wanted to measure the length of a lake?
Can you stretch a tape measure over the lake? (Students may answer about a laser measuring device, which you may want to point out how expensive something
like that is.)
Could you measure lengths along the shore of that lake? (with a long enough tape measure, or some other device, yes)
If you could measure two straight lengths that connect at a 90 degree angle and have end points on opposite ends of the lake, what shape would you form? (A
picture may be helpful, answer should be right triangle)
Well, sometimes it is easy to measure two lengths of a right triangle, and not so easy to measure the third length. Using the Pythagorean Theorem, we could calculate
the third length.
Let's say I wanted to know the length of the diagonal of a standard sheet of loose leaf paper. What are the dimensions of a standard piece of loose leaf? (Some
students may point out that many loose leaf manufacturers have produced loose leaf in 8 x 10.5 inch sizes. You could use this as a teaching moment to point out how
manufacturers love to make things smaller and pass them off as the same but with a new lower price, or you could just say computer paper instead, which still
measures 8.5 x 11 inches.)
Could I use the Pythagorean to find the length of that diagonal? (A picture might be helpful, draw the diagonal in, if they don't see the right triangle connection. Try
and get them to give you the sides as the legs.)
If one leg is 8.5 inches and the other leg is 11 inches, what do I do to figure out what the diagonal is? (Some students may incorrectly respond - "Add them together" Although that would result in a length longer than either of the two legs, I don't think the diagonal is that long)
What did we do with the lengths of the triangle before? (Refer them to the picture with the squares.)
What do we do with those squares? (add them together 72.25 + 121= 193.25 )
Is this the length of the diagonal? (does the paper look like it has a diagonal of 193.25 inches?)
Let's look at the formula. a2+b2=c2
Try and get the students to recognize that they have calculated the left hand side of the formula and that it is equal to c squared, not just c.
How do we figure out what c is, if c squared is 193.25? (divide it by 2? - that would make it smaller, but is 96.625 a reasonable length for the paper's diagonal?)
How do we undo a square? (square root)
page 2 of 5 Have a student use a calculator to get the answer of 13.901 (rounded to the nearest hundredth)
Go back to the original 3-4-5 triangle.
What if we knew the bottom was a length of 3, and the hypotenuse was a length of 5, but did not know what the other side was?
Let's start with the formula a2+b2=c2
Where do the 3 and the 5 go in this equation? (a and b? - One of them is going to be one of those, but remember the longest side is the hypothesis, c. Looking for a
and c or b and c)
Does it matter if the 3 goes in for a or b?(No, the commutative property of addition)
How do we solve from there?
What are the three basic steps to solving a Pythagorean Theorem problem.
Step 1: Write the equation. Most students will start by writing the numbers in the equation, but it is a good idea to have the students get in the habit of writing out
formulas first.
Step 2: Substitute the length of the hypotenuse for c and any lengths of the legs for either a or b. You could just say substitute known values, but some additional
detail doesn't hurt here. Some students might respond better to using the term "Plug-in" instead of substitute, but you want them to get used to the higher
vocabulary.
Step 3: Solve the equation for the missing side.
Some lower level students might appreciate more detailed steps. This is a judgement call, since too many steps often scare the students. However, if you think they
would benefit from more detailed steps, you could use the following:
Step 1: Write the equation
Step 2: Identify the hypotenuse
Step 3: Substitute or "Plug-in" the value for the hypotenuse for c and the values for the legs for a and b.
Step 4: Simplify the Exponents
Step 5: Solve the equation for the remaining squared variable
Step 6: Calculate the square root to get the length of the missing side
Technically, this all boils down to what I like to call a simple "plug'n'chug" (Substitute and evaluate an expression or equation for given variable values) operation, but
some students won't see it that way.
Students will generally understand how to solve when they are given the legs and need the hypotenuse, but not when they are given the hypotenuse and have to find
one of the legs this throws them off. You'll want to start with problems that have the two legs given, and then show them one where the hypotenuse is given and ask
the following:
Should I add these two together?
Someone is bound to say yes. Remind them that the numbers are on opposite sides of the equal sign and you are solving for a leg. To keep an equation balanced,
whatever you add to one side must be added to the other.
Someone is bound to do a2-c2 instead of c2-a2. Ask students how they can be sure which of these will yield the correct answer. Ask them to try this with the 3-4-5
triangle to see if they get ans answer that makes sence.
Have the students work on the Pythagorean Theorem Worksheet. This can be started in class as either individual work, or group work for accommodations. It can also
be used as a homework assignment. The answers could be posted as the following days Bell Ringers with the note "Check your Homework Answers", or can be
provided at the end of class for a student self check.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Guided Practice can be achieved through the use of the Pythagorean Theorem Worksheet in class, or through the use of individual white boards. With individual white
boards, you can use the examples in the worksheet or from other sources, or make up some of your own. It is usually easier to work out the answers for questions
that utilize one of the Pythagorean Triples. You can find a list of these at Math is Fun Pythagorean Triples Listing. You will have to scroll down the page to view the
listing.
To use iIndividual whiteboards, have some students pass out a small whiteboard, one or two dry erase markers, a rag or some other dry marker eraser to each
student. You write a problem on the board and the students work out the problem on the individual whiteboards. Students hold up their boards when completed, and
you give them a thumbs up or thumbs down for their answer. It is usually a good idea to say something like "look at your length for a, does that seem like a
reasonable answer? Isn't that bigger than the hypotenuse? We're looking for the hypotenuse, is your answer bigger than the legs?" Of course this is only possible, if
the students show their work on the boards.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Independent practice can be achieved through the use of the Pythagorean Theorem Worksheet as homework, or through the use of individual whiteboards as
discussed in the guided practice section.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Review what students accomplished during the lesson. Remind the students that this right angle relationship is used in the real world for finding unknown
measurements, when a right triangle can be identified. Many occupations require calculating measurements when using a measuring tape is not an option, like finding
the distance across a lake. Promote discussion as to why it is a useful tool that they will want to hold onto. Perhaps have an exit slip asking them to think of one way
they could use the Pythagorean theorem in the real world.
page 3 of 5 Summative Assessment
Part 2 of this lesson includes the student creation of a net drawing with some calculations of the missing side of a triangle. This could be used as an assessment, or
you could use the Pythagorean Theorem Quiz Students will need a basic calculator with square root capability.
Formative Assessment
You may want to provide a bell ringer the day of this lesson involving the solving of a two-step equation and/or evaluating an equation with a structure similar to the
Pythagorean Theorem. This will help assess prior knowledge. Some Examples:
Solve for x:
1. x2+7=43 (ans: x=±6)
2. 64+x2=164 (ans: x=±10)
Evaluate for a = 12 and b = 5, and c = 13:
1. a2+b2 (ans: 169)
2. c2-b2 (ans: 144)
Students will be provided a Pythagorean Theorem Worksheet with several Pythagorean Theorem questions. They will need to calculate the length of the missing side
using the Pythagorean Theorem. The worksheet includes questions that have missing legs as well as missing hypotenuses. Use student responses to adjust further
instruction, as needed.
Feedback to Students
While students work on the assignment in pairs, groups or individually, the teacher will circulate the room to provide feedback. The teacher will check the accuracy of
their work and ask guiding questions. Students will be provided with the answers to the Pythagorean Theorem Worksheet to check their work, either at the end of the
lesson, or the following day.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Allow students to complete just the right hand side of the worksheet, or just the even numbered ones. Make the Pythagorean Theorem Follow up PowerPoint
available. It works through the problems with the steps broken down a bit further than they had in the original PowerPoint. Allow students to use calculators. Have
students work with partners of mixed abilities. This can often be easily achieved if using a printout of the students' most recent grades in order from highest to lowest.
Figure out how many groups you want. Write the numbers 1-5 or however many groups you want starting from the top down, and then from the bottom up. Distribute
copies of the PowerPoint for those who may be color blind or have trouble seeing the board. It's also a good idea to have a hard copy available for absent students.
If you have time, you may start by getting the students to identify the legs and hypoteneuse, "a, b & c," of a right triangle and how the lengths would be inserted into
the equation, before worrying about how to solve the problems.
Extensions:
Explore the use of the Pythagorean Theorem and its converse to see if the measurements of three segments can form a right triangle. This lesson lays the ground
work for the Origami Boat lesson Resource ID# 49055
Suggested Technology: Computer for Presenter, Basic Calculators, LCD Projector
Special Materials Needed:
Although there is a PowerPoint Presentation in the lesson, it is not required in the delivery of this lesson plan and it can be viewed without the use of Microsoft Office.
There is a free PowerPoint Viewer available through Microsoft that can be used to view the file too.
Further Recommendations:
This is a foundational skills lesson that is meant to lay the ground work for lesson 2 of the Pythagorean Theorem using Origami Boats (resource ID# 49055 Origami
Boats - Pythagorean Theorem in the real world". The next activity will ask the students to create a net drawing of a boat with triangular edges and use the
Pythagorean Theorem to label the lengths on the drawing.
Additional Information/Instructions
By Author/Submitter
This lesson can be run with or without the PowerPoint. If the PowerPoint is used, slide 4 really needs to be explored in more detail, using the questions and details provided
in the non-PowerPoint details.
See resource ID 49055 Origami Boat - Pythagorean Theorem in the real world for a fun second lesson.
This resource is likely to support student engagement in the following the Mathematical Practice: MAFS.K12.MP.8.1 Look for and express regularity in repeated reasoning
SOURCE AND ACCESS INFORMATION
Contributed by: Matthew Funke
Name of Author/Source: Matthew Funke
District/Organization of Contributor(s): Brevard
Is this Resource freely Available? Yes
Access Privileges: Public
page 4 of 5 License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.8.G.2.7:
Description
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical
problems in two and three dimensions.
Remarks/Examples:
Examples of Opportunities for In-Depth Focus
The Pythagorean theorem is useful in practical problems, relates to grade-level work in irrational numbers and plays
an important role mathematically in coordinate geometry in high school.
page 5 of 5